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**Can "infinity" be defined rigorously?**

In math, many of the most fundamental theorems and structures rely on the concept of an infinity.

For example, we define the irrational (and transcendental) numbers as being the limits of some Cauchy sequence of rational numbers. This limiting process inherently assumes that you can continue the sequence "to infinity", but it seems to me we never really quite define exactly what we mean by doing that. The idea of an integral is another example, which is essentially an infinite series.

The number zero, which is sort of like infinity has algebraic properties. When we want to know what we mean by "zero", we can define it as being the element of some set such that,

[itex] \{ x \in S: x + \phi = x \forall x \} , \{ x, \phi \in S: x*\phi = \phi \forall x \} [/itex]

However, as far as I know, infinity has no algebraic properties. It's simply something that sequences tend towards when they fail to converge. Yet, we seem to treat it concretely when actually looking at the limits of such sequences.

It seems like it's almost used to mean different things, depending on the context. What is the rigorous foundation for the idea of infinity? Why should you even be able to take a limit in the first place? Is this simply just a postulate of mathematics?